Simple homotopy theory for Fukaya categories

Abstract

We develop a categorical framework for simple homotopy theory in Fukaya categories, based on the fundamental group of the ambient symplectic manifold. When the first Chern class vanishes, we show that any isomorphism in the Fukaya category of a Weinstein manifold has trivial Whitehead torsion. As an application, we prove that any pair of closed connected Lagrangians that are isomorphic in the Fukaya category of such Weinstein manifolds are simple homotopy equivalent, provided one of the Lagrangians is homotopy equivalent to the ambient symplectic manifold and their fundamental groups are isomorphic.